Land Surface Temperature (LST) Retrieval from GOES Satellite Observations - Satellite-based Applications on Climate Change - page 304

R i ε i B i Tð ð 1 k i W sec θÞþk i sec θWB i TðÞ (19.48)

The Planck function can be expanded into a Taylor series about the brightness
temperature T i

in the form of

DB
DT T i

BT i ðÞ
DB
DT T i

DB
DT T j LTðÞ

R i ¼ B i TðÞ¼ j ¼

j

DB
DT T i T s T

DB
DT T i

B i TðÞB i TðÞþ j ð Þ ¼ j ð T s T i þLTðÞ Þ

DB
DT T i T a T i

DB
DT T i

B i TðÞB i TðÞþ j ð Þ ¼ j ð T a T i þLTðÞ Þ ð 19 : 49 Þ

Inserting Eq. 19.49 into Eq. 19.48 will linearize the RTE with respect to
temperature:

LTðÞε i 1 ð k i w sec θ Þ T s T i þLTðÞ ð Þ þ k i w sec θ T a T i þLTðÞ ð Þ (19.50)

Several approximations have been proposed for L ( T i ). Sun and Pinker ( 2003 )
use

LTðÞT i =n i (19.51)

By inserting Eq. 19.51 into Eq. 19.50 ,

ð C i 1 T i ε i T s Þ ¼ T a ε i T s C i 2 T i ð Þ k i W sec θ (19.52a)

where

1 þ n i ð 1 Þε i ð n i 1 Þ ð 1 ε i Þ

C i 1 ¼ ; C i 2 ¼ (19.52b)

n i n i

Let i represent the 11.0- μ m channel. For most land surfaces and the ocean, the

emissivity at 11.0 m is essentially unity.

In order to reduce the number of unknown variables, we assume that the
atmospheric mean temperature T a is proportional to the surface temperature T s :

μ

T a a w T s (19.53)

It needs to be stated that assumption ( 19.53 ) may introduce errors if the surface
emissivity at 11.0-

μ m channel is not close to unity. A solution for T s can be obtained

as follows:

T i T 11

cW sec

T s

¼

(19.54)

½ ð a w 1 Þk i W sec θ þ 1 θ þ 1